# When was zero actually introduced in mathematics?

Children learn counting things, naturally like, 1, 2, 3, ... and so on. Because it seems obvious to them. But, zero is something we need to teach them about. As far as my understanding goes zero was introduced after quite some time since the birth of mathematics as a science.

But, not sure when and where?

• This is a similar question, but I don't think it's a duplicate. I really like this one; I'm surprised it hasn't been asked before. Nov 6, 2014 at 17:31
• @HDE226868 that's really informative. Thanks. Nov 6, 2014 at 18:49
• Amir Aczel has a new book "Finding Zero" us.macmillan.com/findingzero/amirdaczel. There he finds zero on an inscription from 7th century Cambodia. Perhaps this is earlier than in India. Jan 11, 2015 at 17:49
• Do you mean that they(Cambodians) were using it as a number for mathematical calculations ? or just as a placeholder as some of the other ancient civilizations did. Jan 12, 2015 at 17:38

TL;DR: The modern zero was born in India, in the latter half of the first millennium.

I'll start with Wikipedia, and then work with some better sources. But first, from the Wikipedia article,

Ancient Egyptian numerals were base 10. They used hieroglyphs for the digits and were not positional. By 1740 BCE, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids and distances were measured relative to the base line as being above or below this line.

But that's not really what zero is used for today. Let's move a little.

By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted as a placeholder in the same Babylonian system.

However,

The Babylonian placeholder was not a true zero because it was not used alone.

Moving on a millennium or two, we go to India.

The concept of zero as a number and not merely a symbol or an empty space for separation is attributed to India, where, by the 9th century AD, practical calculations were carried out using zero, which was treated like any other number, even in case of division. The Indian scholar Pingala (circa 5th–2nd century BC) used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), making it similar to Morse code. He and his contemporary Indian scholars used the Sanskrit word śūnya to refer to zero or void.

But perhaps most importantly of all,

The rules governing the use of zero appeared for the first time in Brahmagupta's book Brahmasputha Siddhanta (The Opening of the Universe), written in 628 AD.

Note, however, that he used some concepts little differently:

In some instances, his rules differ from the modern standard.

One key example of this is that for him, $\frac{0}{0}=0$ that is, $0$ divided by $0$ is $0$.

So Wikipedia indicates it was sometime in the 800s that zero got its start.

I'd rather not use Wikipedia as my main source here, so I'll now go elsewhere.

This site confirms that India was possibly the birthplace of the modern zero, and also confirms Brahmagupta's rules:

Brahmagupta, around 650 AD, was the first to formalize arithmetic operations using zero. He used dots underneath numbers to indicate a zero. These dots were alternately referred to as 'sunya', which means empty, or 'kha', which means place. Brahmagupta wrote standard rules for reaching zero through addition and subtraction as well as the results of operations with zero. The only error in his rules was division by zero, which would have to wait for Isaac Newton and G.W. Leibniz to tackle.

It goes on to explain that the concept of zero was studied more extensively in Europe hundreds of years later.

Note: This is a good outline of Brahmagupta, his life, and his work.

Here, we have the a fact listed that Indians used zero in the 5th century:

It began to take shape as a number, rather than a punctuation mark between numbers, in India in the fifth century A.D., says Robert Kaplan, author of The Nothing That Is: A Natural History of Zero (Oxford University Press, 2000). "It isn't until then, and not even fully then, that zero gets full citizenship in the republic of numbers," Kaplan says.

And in the teaser for Kaplan's book, we find

We follow the trail to the East where, a millennium or two ago, Indian mathematicians took another crucial step. By treating zero for the first time like any other number, instead of a unique symbol, they allowed huge new leaps forward in computation, and also in our understanding of how mathematics itself works.

The consensus of these three sources is that while ancient cultures used some concept of "nothing" or a "placeholder", the idea of 0 as a number can be traced back to India, sometime in the second half of the first millennium. I'd recommend sifting through here, possibly the most comprehensive history of 0 I was able to find. It agrees with all the information on India presented above, and talks about one of the earliest inscriptions utilizing 0:

We have an inscription on a stone tablet which contains a date which translates to 876. The inscription concerns the town of Gwalior, 400 km south of Delhi, where they planted a garden 187 by 270 hastas which would produce enough flowers to allow 50 garlands per day to be given to the local temple. Both of the numbers 270 and 50 are denoted almost as they appear today although the 0 is smaller and slightly raised.

The usage here seems to be more of a placeholder, though.

By the way, the answers to this question agree with the India-was-first claim; one uses the Yale source I used here.

• So, if we want to be more precise, are we giving this to Indian mathematician Brahmagupta. Who mentioned rules for zero in his book written in 628 AD. Nov 6, 2014 at 17:58
• I think so. Although the source I mentioned in the last paragraph gave partial credit to a few other Indian mathematicians, Brahmagupta was center-stage. Nov 6, 2014 at 18:00
• Interestingly enough Ptolemy appropriated zero place holder from Babylonians and used O symbol for it. "Ptolemy is using the symbol both between digits and at the end of a number and one might be tempted to believe that at least zero as an empty place holder had firmly arrived. This, however, is far from what happened. Only a few exceptional astronomers used the notation and it would fall out of use several more times before finally establishing itself." math.harvard.edu/~engelwar/MathE300/A%20history%20of%20Zero.pdf Nov 6, 2014 at 23:08
• @Confold Yeah, I had seen something like that. The link's cool, though. Nov 6, 2014 at 23:09
• @HDE226868 No biggie but in the sentence "Here, we have the a fact listed that Indians used zero in the 5th century:", the link on the word "Here" point to this very page which I'm guessing was unintentional. Nov 18, 2014 at 9:26

I went to elementary school in the US. There we learned that the number zero was invented in Central America ...

HERE is an example with that location included.

• Your linked page basically says that the Mayans had a zero placeholder using a shell shape, but "[d]espite the use of zero in the place value system, it was never used for calculations." The Babylonians had already gotten that far. Do we really want to call this the introduction of zero? Dec 31, 2014 at 21:36
• Only the OP knows what he intended by the question. Dec 31, 2014 at 21:39
• I was basically interested in both a) as a placeholder b) as a number in calculations Dec 31, 2014 at 22:48

Concept of Shunya (Zero) The concept of Shunya, or zero void, was originally conceived as the symbol of Brahman, expressing the sum of all distinct forms. The symbol of zero and the decimal system of notation is described in the Atharvaveda[15]. it describes how the number increases by 10 by writing zero in front of it. While there is no explicit mention of zero, it must have been common knowledge based on how it is used.

In fact, the concept of shunya was not just mathematical or scientific, but is deeply rooted in all branches of thought - especially metaphysics and cosmology. Shunya is the transition point between oposites, it symboliss the real balance between divergent tendencies. Most ancient mathematicians defined zero as the sum of two equal and opposite quantities. Zero produces all figures, but is itself not limited to a certain value. Zero is the primary or final reservoir of all single numbers. The symbol of zero and the decimal system of notation is described in the Atharvaveda[16]. It describes how the number increases by 10 by writing zero in front of it. It is said to be described in Atharva Veda. http://www.hindupedia.com/en/Mathematics_of_the_Vedas#Atharva_Veda